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I'm trying to find positive integer solutions to the ellipse $$x^2 - xy + y^2 - k^2 = 0$$ where $k$ is a constant. Specifically, I already have two solutions for a given $k$, and I'm trying to find a third, possibly by using the two known solutions. (I'm trying to copy the technique of producing rational solutions to a conic from one known rational point on the conic.)

I have searched a lot on the Internet, but most resources either suggest checking all values within the range of the ellipse's 'box', or give methods for a particular type of ellipse. Edit: I found some questions which are similar to mine, but I cannot apply the technique used in them to my question mainly because I do not understand the technique. They all mention work by Fricke and Klein (1897).

My questions are:

  • How many [positive] integer solutions does a general ellipse have?
  • How can we find them? (From scratch, or knowing a few solutions beforehand?)
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  • $\begingroup$ You'll find useful information in answers to this question. $\endgroup$ – Ghartal Jun 23 '16 at 10:56
  • $\begingroup$ @Ghartal That question, and many other linked questions, are similar to mine but since none of them tell me what the "Fricke and Klein (1897)" method/theorem is, I cannot apply the same technique here. $\endgroup$ – shardulc says Reinstate Monica Jun 23 '16 at 11:25
  • $\begingroup$ artofproblemsolving.com/community/c3046h1048219 $\endgroup$ – individ Jun 23 '16 at 12:00
  • $\begingroup$ @individ I am having a little difficulty in understanding the method. What do you mean by "if the root of the whole", "if a root $\sqrt{fa}$", etc.? $\endgroup$ – shardulc says Reinstate Monica Jun 23 '16 at 12:18
  • $\begingroup$ In this case. $\sqrt{fa}=k$ It is possible to factorize and to consider all solutions. $\endgroup$ – individ Jun 23 '16 at 12:22
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The answer depends on the number of prime factors of $k$ when we consider a factorization over the ring of Eisenstein integers $\mathbb{Z}[\omega]$. Have a look at this answer, too. For instance, there are just trivial solutions if $k$ is a square-free number and a product of primes of the form $3m-1$, that do not split over $\mathbb{Z}[\omega]$. So the number of integer points on such ellipses has a very irregular arithmetic behaviour, but a quite regular behaviour on average, just like in the Gauss circle problem.

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  • $\begingroup$ Thanks! By the way, the linked question had prompted me to ask this question :) $\endgroup$ – shardulc says Reinstate Monica Jun 24 '16 at 4:55
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The equation $x^2-xy+y^2=k^2$ has a parametric solution given below,

$$x=p^2-q^2$$

$$y=2pq-q^2$$

$$k=p^2-pq+q^2$$

For $(p,q)=(3,2)$ we get $(x,y,k)=(5,8,7)$.

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